Differential Geometry, Spring 2012.

Instructor: Dmitry Ryabogin

Assignment 11.

1. Problem 1. a) Find the normal curvature of the paraboloid z = \frac{1}{2}(ax^{2} + by^{2}) at the point (0, 0) in the direction (dx : dy).

b) Prove that for any parametrization of the plane the second quadratic form of the plane is zero; the second quadratic form of the sphere is proportional to the first one.

c) Find the asymptotic curves of the surface z = \frac{x}{y} + \frac{y}{x}.

d) Find the asymptotic curves of the catenoid

r(u, v) = (cosh u cos v, cosh u sin v, u).

e) Prove that on the helicoid

r(u, v) = (u cos v, u sin v, bv), b > 0,

there are two families of the asymptotic curves: the straight lines and helixes.

2. Problem 2. a) Find the main curvatures of the paraboloid z = a (x^{2} + y^{2}) at the point (0, 0, 0).

b) Find the lines of curvature on the helicoid and the paraboloid z = a xy.

c) Find the mean and the Gaussian curvature of the paraboloid z = a xy at the point x = y = 0.

d) Prove that the mean curvature of the helicoid is zero.

e) Prove that the mean curvature of the catenoid z = a arcosh ^{\frac{\sqrt{x^{2}+y^{2}}}{a}} is zero.

3. Problem 3. Prove or give a counterexample:

a) If a curve is both asymptotic curve and a line of curvature, then it must be planar.

b) If a curve is planar and an asymptotic curve, then it must be a line.

4. Problem 4. a) Show that at a hyperbolic point, the principal directions bisect the asymptotic directions

b) Show that if a surface is tangent to a plane along a curve, then the points of this curve are either parabolic or planar.